What to change after changing the paths in the database?

Question

Database

Hello!
Please tell me if you need to look for or fix something after the following operation:
I handed over the site to my hoster for hosting, the hoster said that the links in my database were not written correctly, but they did something and the site began to work ...
After a while, I decided to put the database in order. I requested a backup of the site, installed it in openserver, using a script ( https://interconnectit.com/products/search-and-rep... ) changed all references from "test_hosting.ru" to "my_site.ru"
So far everything is working True, without pictures, but that's another topic... The
question is this:
If the paths were written incorrectly in the database, BUT everything worked as it should, what could the hoster do to solve this problem? Redirect?
If I changed the paths in the database, then there is no need for a redirect anymore? Does that mean it can be removed?
And if it's not a redirect then what?
In other words, what should be done next, after changing the paths in the database? Where to look? What to search?